Prevent dirt from splashing up on my siding?

If you don't want to use gutters you could try one of those strips of metal they put on the roof which direct the water flow to the sides (see the link). If you don't want to do that either, you will have to use stones or pebbles as you suggest. First lay some plastic under the stone area, then put the stones in. You won't be able to grow anything there as any exposed soil will splash.